A physical quantity $z$ depends on four observables $a, c, c$ and $d$, as $z=\frac{a^{2} b^{\frac{2}{3}}}{\sqrt{c} d^{3}}$. The percentages of error in the measurement of $\mathrm{a}, \mathrm{b}, \mathrm{c}$ and d are $2 \%, 1.5 \%, 4 \%$ and $2.5 \%$ respectively. The percentage of error in z is:

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Units & Measurement

A physical quantity

z

depends on four observables

a, c, c

and

d

, as

z = \frac{a ^{2} b ^{\frac{2}{3}}}{c d ^{3}}

. The percentages of error in the measurement of

a, b, c

and d are

2%, 1.5%, 4%

and

2.5%

respectively. The percentage of error in z is:

Answer: (D)

Because

z = \frac{a ^{2} b ^{2/3}}{c d ^{3}}

is a product of powers, the fractional errors add, each weighted by the magnitude of its exponent:

\frac{Δ z}{z} = 2 \frac{Δ a}{a} + \frac{2}{3} \frac{Δ b}{b} + \frac{1}{2} \frac{Δ c}{c} + 3 \frac{Δ d}{d}

. Inserting the percentage errors,

2 (2%) + \frac{2}{3} (1.5%) + \frac{1}{2} (4%) + 3 (2.5%) = 4% + 1% + 2% + 7.5% = 14.5%

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option